Totally Ordered Set/Examples/Example Ordering on Integers

Examples of Totally Ordered Sets

Let $\preccurlyeq$ denote the relation on the set of integers $\Z$ defined as:

$a \preccurlyeq b$ if and only if $0 \le a \le b \text { or } b \le a < 0 \text { or } a < 0 \le b$

where $\le$ is the usual ordering on $\Z$.

Then $\struct {\Z, \preccurlyeq}$ is a totally ordered set.

Proof

From Example Ordering on Integers, $\preccurlyeq$ is an ordering on $\Z$.

Let $a, b \in \Z$.

Let $a, b \ge 0$.

Then either $a \le b$ or $b \le a$.

Hence either $a \preccurlyeq b$ or $b \preccurlyeq a$.

Let $a, b < 0$.

Then either $a \le b$ or $b \le a$.

Hence either $a \preccurlyeq b$ or $b \preccurlyeq a$.

Let $a < 0$ but $b \ge 0$.

Then $a \preccurlyeq b$.

Let $b < 0$ but $a \ge 0$.

Then $b \preccurlyeq a$.

In all cases, either $a \preccurlyeq b$ or $b \preccurlyeq a$.

Hence the result.

$\blacksquare$